Transmit diversity is considered to be important since it can be applied at transmitter side to improve the reliability of communication link even if channel state information is not available at the transmitter. That Includes high speed mobility scenarios where feedback information from receiver to transmitter becomes quickly obsolete and also broadcast scenarios.
Alamouti code, which is described in S. M. Alamouti: “A simple transmit diversity technique for wireless communications”, IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, October 1998, is a popular transmit diversity scheme from orthogonal design for two transmit antennas since it has the following desired properties of:                1. Full diversity        2. Full rate (no rate loss)        3. Simple linear receiver, i.e. the processing required at the receiver scales linearly in the number of transmit antennas.        
Alamouti code has been widely adopted, e.g. for LTE OFDM systems.
However, the direct application of Alamouti scheme for FBMC/OQAM is not possible So far, considerable amount of efforts have been spent by many industries and universities to find competitive solution for transmit diversity with FBMC/OQAM, but all the state-of-the-art solutions have some drawbacks as explained later.
Before explaining the problem of achieving transmit diversity for FBMC/OQAM, the Alamouti transmission scheme as applied in LTE should be reviewed as illustrated in FIG. 2.
FIG. 2 shows a transmission scheme based on OFDM, wherein the complex QAM symbols s1 and s2 are transmitted from the first transmit antenna Tx1 using the two resources (m0,n0) and (m0+1,n0) where m0 and n0 denote subcarrier and time indices, respectively. From the second transmit antenna Tx2 their complex conjugate versions of the QAM symbols are transmitted with or without taking its minus by using the same two resources, but now the used resources for s1 and s2 are exchanged. If it is assumed that the complex-valued channel frequency responses on subcarriers m0and m0+1 of symbol n0 are quasi equivalent, denoted as H(1) and H(2) for transmit antennas Tx1 and Tx2, respectively, one can write the received signals for the subcarriers m0 and m0+1 as shown in equations (1) and (2):ym0,n0=H(1)s1−H(2)s*2+ηm0,n0  (1)ym0+1,n0=H(1)s2+H(2)s*1+ηm0+1,n0  (2)where ηm0,n0 is an AWGN. With some arrangement of these receive signals, the following linear equation system is obtained:
                              [                                                                      y                                                            m                      0                                        ,                                          n                      0                                                                                                                                            y                                                                                    m                        0                                            +                      1                                        ,                                          n                      0                                                        *                                                              ]                =                                                                              [                                                                                                              H                                                      (                            1                            )                                                                                                                                                -                                                      H                                                          (                              2                              )                                                                                                                                                                                                                    H                                                                                    (                              2                              )                                                        *                                                                                                                                                H                                                                                    (                              1                              )                                                        *                                                                                                                                ]                                ⁡                                  [                                                                                                              s                          1                                                                                                                                                              s                          2                          *                                                                                                      ]                                            +                              [                                                                                                    η                                                                              m                            0                                                    ,                                                      n                            0                                                                                                                                                                                                  η                                                                                                            m                              0                                                        +                            1                                                    ,                                                      n                            0                                                                          *                                                                                            ]                                      →            y                    =                      Hs            +            η                                              (        3        )            
Then, at the receiver the following linear processing is performed:
                              [                    ]                =                                            1                                                                                                              H                                              (                        1                        )                                                                                                  2                                +                                                                                                H                                              (                        2                        )                                                                                                  2                                                      ⁢                          H              H                        ⁢            y                    =                                    [                                                                                          s                      1                                                                                                                                  s                      2                      *                                                                                  ]                        +                                          1                                                                                                                          H                                                  (                          1                          )                                                                                                            2                                    +                                                                                                          H                                                  (                          2                          )                                                                                                            2                                                              ⁢                              H                H                            ⁢                              η                .                                                                        (        4        )            
The diversity order of 2 is achieved assuming that the channel response from the transmit antennas Tx1 and Tx2 are Independent.
To summarize, the main idea of Alamouti coding is that it is an orthogonal design since only linear combination is needed and it incurs no rate loss because two resources are utilized to deliver two data symbols.
Now, fundamental properties of FBMC/OQAM that are essential for understanding why the Alamouti scheme cannot be directly applied to FBMC/OQAM should be reviewed on the basis of FIG. 3. In the single antenna, single input, single output (SISO) system shown in FIG. 3, a real-valued pulse amplitude modulation (PAM) signal am0,n0 is transmitted using the resource (m0,n0).
The respective baseband equivalent receive signal may be written as
                                          y                                          m                0                            ,                              n                0                                              =                                                    H                                                      m                    0                                    ,                                      n                    0                                                              ⁡                              (                                                      a                                                                  m                        0                                            ,                                              n                        0                                                                              +                                      j                    ⁢                                                                                  ⁢                                          I                                                                        m                          0                                                ,                                                  n                          0                                                                                                                    )                                      +                          η                                                m                  0                                ,                                  n                  0                                                                    ⁢                                  ⁢        where                            (        5        )                                          I                                    m              0                        ,                          n              0                                      =                              ∑                                                            (                                      p                    ,                    q                                    )                                ≠                                  (                                      0                    ,                    0                                    )                                                            p                ,                                  q                  ∈                                      {                                                                  -                        1                                            ,                      0                      ,                                              +                        1                                                              }                                                                                ⁢                                    a                                                                    m                    0                                    +                  p                                ,                                                      n                    0                                    +                  q                                                      ⁢                                          〈                g                〉                                                                                  m                    0                                    +                  p                                ,                                                      n                    0                                    +                  q                                                                                        (        6        )            
is the so called intrinsic interference coming from data symbols on neighbor subcarriers and symbols. The coefficients gm0+p,n0+q are called ambiguity function that captures the characteristic of the used prototype filter. Here, it is assumed that a good localized filter is used such that the intrinsic interference is caused only by the immediate neighbor resources, but in general other resources that are located farther apart could also contribute to form the intrinsic interference.
It can be seen that, unlike OFDM, the subcarrier signal is not orthogonal in the complex domain. It is, however, possible to restore the orthogonality in the real domain by channel equalization and taking its real part as
                                          a            ^                                              m              0                        ,                          n              0                                      =                              Re            ⁢                          {                                                y                                                            m                      0                                        ,                                          n                      0                                                                                        H                                                            m                      0                                        ,                                          n                      0                                                                                  }                                ≈                                    a                                                m                  0                                ,                                  n                  0                                                      +                          η                                                m                  0                                ,                                  n                  0                                            ′                                                          (        7        )            
As it can be seen next, there is the consequence on the transmit diversity from the fact that FBMC/OQAM loses the complex orthogonality.
For transmit diversity for FBMC/OQAM, the system model shown in FIG. 4 is considered.
In the scenario shown in FIG. 4, two real-valued PAM signals a1 and a2 are transmitted using the two resources (m0,n0) and (m0+1,n0) from the transmit antenna Tx1. From transmit antenna Tx2, these PAM symbols are transmitted with and respectively without taking its minus by using the same two resources but now the used resources for a1 and a2 are exchanged as can be seen in FIG. 4. With the same assumptions on H(1) and H(2) from transmit antennas Tx1 and Tx2, the receive signals for the subcarriers m0 and m0+1 read as
                              y                                    m              0                        ,                          n              0                                      =                                                            H                                  (                  1                  )                                            ⁢                                                (                                                            a                      1                                        +                                          j                      ⁢                                                                                          ⁢                                              I                                                                              m                            0                                                    ,                                                      n                            0                                                                                                    (                          1                          )                                                                                                      )                                                  ︸                                                            =                      Δ                                        ⁢                                                                                  ⁢                                          s                      1                                                                                            -                                          H                                  (                  2                  )                                            ⁢                                                (                                                            a                      2                                        -                                          j                      ⁢                                                                                          ⁢                                              I                                                                              m                            0                                                    ,                                                      n                            0                                                                                                    (                          2                          )                                                                                                      )                                                  ︸                                      ≠                                                                                  ⁢                                          s                      2                      *                                                                                            +                          η                                                m                  0                                ,                                  n                  0                                                              =                                                    H                                  (                  1                  )                                            ⁢                              s                1                                      -                                          H                                  (                  2                  )                                            ⁡                              (                                                      s                    2                    *                                    +                                      j                    ⁢                                                                                  ⁢                                          I                      1                                                                      )                                      +                          η                                                m                  0                                ,                                  n                  0                                                                                        (        8        )                                          y                                                    m                0                            +              1                        ,                          n              0                                      =                                                            H                                  (                  1                  )                                            ⁢                                                (                                                            a                      2                                        +                                          j                      ⁢                                                                                          ⁢                                              I                                                                                                            m                              0                                                        +                            1                                                    ,                                                      n                            0                                                                                                    (                          1                          )                                                                                                      )                                                  ︸                                                            =                      Δ                                        ⁢                                                                                  ⁢                                          s                      2                                                                                            +                                          H                                  (                  2                  )                                            ⁢                                                (                                                            a                      1                                        +                                          j                      ⁢                                                                                          ⁢                                              I                                                                                                            m                              0                                                        +                            1                                                    ,                                                      n                            0                                                                                                    (                          2                          )                                                                                                      )                                                  ︸                                      ≠                                                                                  ⁢                                          s                      2                      *                                                                                            +                          η                                                                    m                    0                                    +                  1                                ,                                  n                  0                                                              =                                                    H                                  (                  1                  )                                            ⁢                              s                2                                      +                                          H                                  (                  2                  )                                            ⁡                              (                                                      s                    1                    *                                    +                                      j                    ⁢                                                                                  ⁢                                          I                      2                                                                      )                                      +                          η                                                                    m                    0                                    +                  1                                ,                                  n                  0                                                                                        (        9        )            
where the complex-valued “virtual symbols” s1 and s2 are defined as the real-valued desired signal plus intrinsic interference.
In an attempt to implement the Alamouti scheme, the following is introduced:I1Im0+1,n0(1)−Im0,n0(2)  (10)I2Im0,n0(1)+Im0+1,n0(2)  (11)
With some arrangement of these received signals, the following linear equation system is obtained in a similar way as the Alamouti scheme for OFDM as explained above:
                              [                                                                      y                                                            m                      0                                        ,                                          n                      0                                                                                                                                            y                                                                                    m                        0                                            +                      1                                        ,                                          n                      0                                                        *                                                              ]                =                                                            [                                                                                                    H                                                  (                          1                          )                                                                                                                                    -                                                  H                                                      (                            2                            )                                                                                                                                                                                                  H                                                                              (                            2                            )                                                    *                                                                                                                                    H                                                                              (                            1                            )                                                    *                                                                                                                    ]                                            ︸                                  Orthogonal                  ⁢                                                                          ⁢                  design                                                      ⁡                          [                                                                                          s                      1                                                                                                                                  s                      2                      *                                                                                  ]                                +                      j            ⁢                                          [                                                                                                    -                                                  H                                                      (                            2                            )                                                                                                                                                              I                        1                                                                                                                                                H                                                                              (                            2                            )                                                    *                                                                                                                                    I                        2                                                                                            ]                                            ︸                                  Orthogonality                  ⁢                                                                          ⁢                  is                  ⁢                                                                          ⁢                  lost                                                              +                      [                                                                                η                                                                  m                        0                                            ,                                              n                        0                                                                                                                                                              η                                                                                            m                          0                                                +                        1                                            ,                                              n                        0                                                              *                                                                        ]                                              (        12        )            
It can however be observed that the orthogonality is lost due to the second term on the right hand side of the equation system. The main reason for this is that the intrinsic interferences for the transmitted signals from different antennas are not equivalent since the surrounding data of each time-frequency resource grid are different due to the random nature of data signals. This explains the problem that the transmit diversity from the orthogonal design following the Alamouti coding scheme cannot be applied to FBMC/OQAM in a straightforward manner.
Several attempts to address this problem of non-orthogonality can be found in the literature and will be discussed in the following:
In M. Bellanger, “Transmit diversity in multicarrier transmission using OQAM modulation,” in Proc. The 3rd Int. Symposium on Wireless Pervasive Computing (ISWPC'08), pp. 727-730, May 2008, the author proposes a simple delay diversity where no effort is made to realize orthogonality. Although this approach does not have any rate loss, due to its non-orthogonality, it requires very complex maximum likelihood receiver and it does not achieve full diversity.
The authors in H. Lin, C. Lele, P. Siohan, “A pseudo Alamouti transceiver design for OFDM/OQAM modulation with cyclic prefix,” in Proc. SPAWC, 2009 propose to introduce a cyclic prefix which is common for OFDM, but not for FBMC/OQAM.
Because of the cyclic prefix, the orthogonality can be realized, but it results in a rate loss.
Another approach, presented in by C. Lele, P. Siohan, R. Legouable, “The Alamouti scheme with CDMA-OFDM/OQAM”, EURASIP Journal on Advances in Signal Processing, 2010, suggests the spreading and dispreading using Walsh-Hadamard codes for nullifying intrinsic interferences. Thanks to spreading/dispreading, orthogonality can be achieved, but a rate loss results.
In M. Renfors, T. Ihalainen, T. H. Stitz, “A Block-Alamouti Scheme for Filter Bank Based Multicarrier Transmission,” Proceedings of the European Wireless Conference 2010, a block Alamouti scheme using some zero symbols is introduced based on orthogonal design. The idea is to apply the Alamouti scheme to two areas of symbols instead of two symbols such that the intrinsic interference caused from 2 different transmit antennas are equivalent. Some zero symbols are added around the areas to avoid “edge effect”. The zero symbols lead to rate loss. Besides, the applicability of the scheme may be limited since channel has to be constant over blocks that may not hold for many propagation scenarios, e.g. for mobile scenarios.
To summarize, there has not been a solution in the literature that can realize the orthogonal design without rate loss.